Which of the following has an Euler path, but not an Euler circuit?


Question 11Select one:

a. 

K6


b. 

K7


c. 

K3,4


d. 

K2,5




An **Euler path** is a path that visits every edge of a graph exactly once, while an **Euler circuit** is an Euler path that starts and ends at the same vertex. To determine which graph has an **Euler path but not an Euler circuit**, we need to check the necessary conditions:

### Euler Path Conditions:
- A graph has an Euler path if and only if it has **exactly two vertices of odd degree**, while all others have even degrees.

### Euler Circuit Conditions:
- A graph has an Euler circuit if all vertices have an **even degree**.

Now, let's analyze the given options:

#### a. **K6** (Complete graph with 6 vertices)
- In a complete graph \( K_n \), each vertex has a degree of \( n-1 \).
- For \( K_6 \), each vertex has degree \( 6-1 = 5 \), which is odd.
- Since all vertices are odd, **K6 does not have an Euler path or circuit**.

#### b. **K7** (Complete graph with 7 vertices)
- For \( K_7 \), each vertex has degree \( 7-1 = 6 \), which is even.
- Since all vertices have even degrees, **K7 has an Euler circuit** but no Euler path.

#### c. **K3,4** (Complete bipartite graph with partitions of 3 and 4 vertices)
- In \( K_{3,4} \), vertices in the first set (3 vertices) each have degree 4, and vertices in the second set (4 vertices) each have degree 3.
- Thus, the vertices in the first set have even degrees, and the vertices in the second set have odd degrees.
- There are exactly two vertices with odd degrees (from the second set), so **K3,4 has an Euler path but not an Euler circuit**.

#### d. **K2,5** (Complete bipartite graph with partitions of 2 and 5 vertices)
- In \( K_{2,5} \), vertices in the first set (2 vertices) each have degree 5, and vertices in the second set (5 vertices) each have degree 2.
- Here, the vertices in the first set have odd degrees, and the vertices in the second set have even degrees.
- Since there are two vertices with odd degrees (from the first set), **K2,5 has an Euler path but not an Euler circuit**.

### Conclusion:
Both **\( K_{3,4} \) and \( K_{2,5} \)** have Euler paths but not Euler circuits. Therefore, the correct answer to this specific multiple-choice question is:

**c. \( K_{3,4} \)**.

Would you like further clarification on Euler paths and circuits, or to explore related problems? Here are five follow-up questions for you:

1. What are the conditions for a graph to have an Euler circuit?
2. Can a graph with more than two odd degree vertices have an Euler path?
3. How does the degree of a vertex relate to Euler paths?
4. What is a Hamiltonian path, and how does it differ from an Euler path?
5. Can a graph with isolated vertices have an Euler path or circuit?

**Tip:** When analyzing Euler paths, always check the degrees of vertices first!

Which of the following has an Euler path, but not an Euler circuit? a. K6 b. K7 c. K3,4 d. K2,5

Explore the conditions for Euler paths and circuits in graph theory using K3,4 and K2,5. This problem focuses on identifying Euler paths in graphs with odd and even vertex degrees, suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation